Magic Squares---At no time do my fingers....

by

Fred Schaal

The kind of magic squares that I am familiar with are the
ones with an odd
number of cells along an edge. These I refer to as "odd
squares". I sometimes think that one could assemble an even
number of
odd squares so as to produce an even magic square- an even
square. What if I were to use
the very fundamental; 3 by 3 square with just the first nine integers
and
assemble them into a 6 by 6 magic
square?

8

1

6

3

5

7

4

9

2

8

1

6

3

5

7

4

9

2

8

1

6

3

5

7

4

9

2

8

1

6

3

5

7

4

9

2

This in indeed an even "magic" square. Each row and column and
diagonal
sum to the same number: the magic number, aka magic constant.
But it is a trivial
case. My kind of magic squares do not repeat numbers. This one
four- peats each
number of the original primitive single digit magic square. This
square is
unacceptable.

I shall endeavor to find out how to construct
even magic
squares! This link contains instructions for constructing a 4
by 4
magic square with the first 16 consecutive counting numbers. The
magic
constant is 43. Here it is....

6

2

3

13

5

11

10

8

9

7

6

12

4

14

15

1

Unfortunately,
when I try to expand it to 6 by 6 or 8 by 8 the method fails .
But at last
and at least I can do a 4 by 4 by a set of rules, that is, systematically.
(At http://www.primepuzzles.net/puzzles/puzz_068.htm
I found a 16 by 16 square that uses the prime numbers from 11 to
2633. Gasp!)

Below please find a
12 by 12 table for a magic square:

This is really an awful magic square. Again I am at
a lost to know from
whence it came. Its members are consecutive prime numbers
starting
with 1. Whoever was involved with its creation truly have too
much time on
their hands.

This page seems to be going nowhere slowly.
By way
of a summary thought: The rules for odd squares do not have a
counterpart
in the land of even squares. It seems that each size has its own
pattern
of algorithm. Naturally I am crushed by this bolt of reality but
I hope to
learn to live with it. I cannot believe that I have not tried a website
from NCTM,
the National Council of Teachers of Mathematics. Is there a
message here?